Largest Repetition Factorization of Fibonacci Words

Abstract

A factorization of a string w is said to be a repetition factorization of w if every factor in the factorization is a repetition (i.e., the factor has a period shorter than or equal to the half of its length). Inoue et al. [TOCS 2022] showed how to compute the largest/smallest repetition factorization of a given string w of length n in $O(n \log n)$ time and O(n) space, by reducing the problems to the longest/shortest path problems on the repetition graph built on w. Inoue et al. also considered repetition factorizations on Fibonacci words, and posed a conjecture on the size $S_{F_k}$ of the largest repetition factorization of the k-th Fibonacci word $F_k$. In this work, we provide a complete proof for this problem, by showing that $S_{F_k}$ is given by the recurrence $S_{F_k} = S_{F_{k-1}} + S_{F_{k-2}} + 1$ for every $k \ge 15$.